| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Rotation about x-axis, region between two curves |
| Difficulty | Standard +0.8 This is a multi-step volumes of revolution problem requiring students to find intersection points, set up the integral for rotation about the x-axis with two curves (requiring subtraction of volumes), and evaluate integrals involving both polynomial and reciprocal functions. The algebraic manipulation and integration of 4/x² is more demanding than standard single-curve rotation problems, placing it above average difficulty. |
| Spec | 1.08d Evaluate definite integrals: between limits4.08d Volumes of revolution: about x and y axes |
4.\\
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The diagram shows the curves $y = ( x - 1 ) ^ { 2 }$ and $y = 2 - \frac { 2 } { x } , x > 0$.\\
(i) Verify that the two curves meet at the points where $x = 1$ and where $x = 2$.
The shaded region bounded by the two curves is rotated completely about the $x$-axis.\\
(ii) Find the exact volume of the solid formed.\\
\hfill \mbox{\textit{OCR C3 Q4 [9]}}